# Mathematical Thought From Ancient To Modern Times

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Publication date
1972

Topics
Mathematics -- History

Collection
opensource

Language
English

[ Number of Pages := 1400 ] 3 Volumes in a Single Book

More Details

Author := Kline, Morris, 1908-1992

Title := Mathematical thought from ancient to modern times.

Imprint := New York : Oxford University Press, 1972.

[ Library of Congress Class Number := QA21 .K53 ]

UofT at Mississauga Stacks QA21 .K53

ISBN := 0195014960:

Catalogue Key := 2795115

Includes bibliographies.

=================================================

☀► Table of Contents ☀►

1. Mathematics in Mesopotamia, 3

1. Where Did Mathematics Begin? 3

2. Political History in Mesopotamia,4

3. The Number Symbols, 5

4. Arithmetic Operations, 7

5. Babylonian Algebra, 8

6. Babylonian Geometry, 10

7. The Uses of Mathematics in Babylonia, I I

8- Evaluation of Babylonian Mathematics, 13

2. Egyptian Mathematics, 15

1. Background, 15

2. The Arithmetic, 16

3. Algebra and Geometry, 18

4. Egyptian Uses of Mathematics, 21

5. Summary, 22

3. The Creation of Classical Greek Mathematics,24

1. Background, 24

2. The General Sources,25

3. The Major Schools of the Classical Period,27

4. The Ionian School, 28

5. The Pythagoreans, 28

6. The Eleatic School, 34

7. The Sophist School, 37

8. The Platonic School, 42

9. The School of Eudoxus, 48

10. Aristotle and His School, 5l

4. Euclid and Apollonius, 56

1. Introduction, 56

2. The Background of Euclid's Elements,5T

3. The Definitions and Axioms of the Elements,SE

4. Books I to IV of the Elements,60

5. Book V: The Theory of Proportion, 68

6. Book VI : Similar Figures, 73

7. Bools VII, VIII, and IX: The Theory of Numbers, 77

8. Book X: The Classification of Incommensurables, 80

9. Books XI, XII, and XIII: Solid Geometry and the Method of Exhaustion, El

10. The Merits and Defects of the Elements, S6

11. Other Mathematical Works by Euclid,88

12. The Mathematical Work of Apollonius, S9

5. The Alexandrian Greek Period: Geometry and Trigonometry, l0l

1. The Founding of Alexandria, l0l

2. The Character of Alexandrian Greek Mathematics, 103

3. Areas and Volumes in the Work of Archimedes, 105

4. Areas and Volumes in the Work of Heron, 116

5. Some Exceptional Curves, ll7

6. The Creation of Trigonometry, I 19

7. Late Alexandrian Activity in Geometry, 126

6. The Alexandrian Period: The Reemergence of Arithmetic and Algebra, 131

1. The Symbols and Operations of Greek Arithmetic, l3l

2. Arithmetic and Algebra as an Independent Development, 135

7. The Greek Rationalization of Nature, 145

1. The Inspiration for Greek Mathematics, 145

2. The Beginnings of a Rational View of Nature, 146

3. The Development of the Belief in Mathematical Design, 147

4. Greek Mathematical Astronomy 154

5. Geography, 160

6. Mechanics, 162

7. Optics, 166 8. Astrology, 168

8. The Demise of the Greek World, 171

1. A Review of the Greek Achievements, l7l

2. The Limitations of Greek Mathematics, 173

3. The Problems Bequeathed by the Greeks, 176

4. The Demise of the Greek Civilization, 177

9. The Mathematics of the Hindus and Arabs, lB3

1. Early Hindu Mathematics, 183

2. Hindu Arithmetic and Algebra of the Period e.o, 20G-1200, 184

3. Hindu Geometry and Trigonometry of the Period e.o, 200-1200, 188

4. The Arabs, 190

5. Arabic Arithmetic and Algebra, l9l

6. Arabic Geometry and Trigonometry, 195

7. Mathematics circa 1300, 197

10. The Medieval Period in Europe, 200

1. The Beginnings of a European Civilization, 200

2. The Materials Available for Learning, 201

3. The Role of Mathematics in Early Medieval Europe, 202

4. The Stagnation in Mathematics, 203

5. The First Revival of the Greek Works, 205

6. The Revival of Rationalism and Interest in Nature, 206

7. Progress in Mathematics Proper, 209

8. Progress in Physical Science, 2ll 9. Summary,2l3

11. The Renaissance, 21 6

1. Revolutionary Influences in Europe, 216

2. The New Intellectual Outlook, 218

3. The Spread of Learning, 220

4. Humanistic Activity in Mathematics, 221

5. The Clamor for the Reform of Science, 223

6. The Rise of Empiricism, 227

12. Mathematical Contributions in the Renaissance, 231

1. Perspective,23l

2. Geometry Proper, 234

3. Algebra, 236

4. Trigonometry, 237

5. The Major Scientific Progress in the Renaissance, 240

6. Remarks on the Renaissance, 247

13. Arithmetic and Algebra in the Sixteenth and Seventeenth Centuries, 250

1. Introduction, 250

2. The Status of the Number System and Arithmetic, 251

3. Symbolism,259

4. The Solution of Third and Fourth Degree Equations,263

5. The Theory of Equations, 270

6. The Binomial Theorem and Allied Topics, 272

7. The Theory of Numbers, 274

8.The Relationship of Algebra to Geometry, 278

14. The Beginnings of Projective Geometry, 285

1. The Rebirth of Geometry, 285

2. The Problems Raised by the Work on Perspective, 286

3. The Work of Desargues, 288

4. The Work of Pascal and La Hire, 295

5. The Emergence of New Principles, 299

15. Coordinate Geometry, 302

1. The Motivation for Coordinate Geometry, 302

2. The Coordinate Geometry of Fermat,303

3.Rend Descartes,SO4

4.Descartes's Work in Coordinate Geometry' 308

5. Seventeenth'Century Extensions of Coordinate Geometry, 317

6' The Importance of Coordinate Geometry 321

16. The Mathematization of Science, 325

1. Introduction, 325

2. Descartes's Concept of Science, 325

3. Galileo's Approach to Science, 327

4. The Function Concept, 335

17. The Creation of the Calculus, 342

1. The Motivation for the Calculus, 342

2. Early Seventeenth-Century Work on the Calculus, 344

3. The Work of Newton, 356

4. The Work of Leibniz, 370

5. A Comparison of the Work of Newton and Leibniz, 378

6. The Controversy over Priority, 380

7. Some Immediate Additions to the Calculus, 381

8. The Soundness of the Calculus, 383

18. Mathematics as of 1700, 391

1. The Transformation of Mathematics, 391

2. Mathematics and Science, 394

3. Communication Among Mathematicians, 396

4. The Prospects for the Eighteenth Century, 398

19. Calculus in the Eighteenth Century, 400

1. Introduction' 400

2. The Function Concept, '103

3. The Technique of Integration and Complex Quantities, 406

4. Elliptic Integrals, 411

5. Further Special Functions' 422

6. The Calculus of Functions of Several Variables, 425

7. The Attempt to Supply Rigor in the Calculus, 426

20. Infinite Series, 436

1. Introduction, 436

2. Initial Work on Infinite Series, 436

3. The Expansion of Functions, 440

4. The Manipulation of Series, 442

5.Trigonometric Series, 454

6. Continued Fractions, 459

7. The Problem of Convergence and Divergence' 460

21. Ordinary Differential Equations in the Eighteenth Century, 468

1. Motivations, 468

2. First Order Ordinary Differential Equations,4Tl

3. Singular Solutions, 476

4. Second Order Equations and the Riccati Equation' 478

5. Higher Order Equations,484

6. The Method of Series, 'i88

7. Systems of Differential Equations, 490 8. Summary,499

22. Partial Differential Equations in the Eighteenth Century, 502

1. introduction 502

2. The Wave Equation, 503

3. Extensions of the Wave Equation' 515

4. Potential Theory, 522

5. First Order Partial Differential Equation' 531

6. Monge and the Theory of Characteristics , 5gO

7. Monge and Nonlinear Second Order Equations, 538

8. Systems of First Order Partial Differential Equations' 5'fl)

9. The Rise of the Mathematical Subject, 542

23. Analytic and Differential Geometry in the Eighteenth Century, 544

1. Introduction, 544

2. Basic Analytic Geometry, 544

3. Higher plane Curves, 547

4. The Beginnings of Differential Geometry, 5.54

5. plane Curves, 555

6. Space Curves, 557

7. The Theory of Surfaces, 562

8. The Mapping problem, 520

24. The Calculus of Variations in the Eighteenth Century, 573

1. The Initial Problems, 573

2.The Early Work of Euler, 577

3.The principle of Least Action, 579

4. The Methodology of Lagrange, 582

5. Lagrange and Least Action,.587

6. The Second Variation, 589

25. Algebra in the Eighteenth Century, 592

1. Status of the Number System, 592

2. The Theory of Equations, 597

3. Determinants and Elimination Theory, 606

4. The Theory of Numbers, 60g

26. Mathematics as of 1800, 614

1. The Rise of Analysis, 614

2. The Motivation for the Eighteenth-Century Work, 616

3. The Problem of Proof 617

4. The Metaphysical Basis,619

5. The Expansion of Mathematical Activity, 621

6. A Glance Ahead, 623

27. Functions of a Complex Variable, 626

1. Introduction, 626

2. The Beginnings of Complex Function Theory,626

3. The Geometrical Representation of Complex Numbers, 628

4. The Foundation of Complex Function Theory, 632

5. Weierstrass's Approach to Function Theory,642

6. Elliptic Functions, 644

7. Hyper Elliptic Integrals and Abel's Theorem, 651

8. Riemann and Multiple-Valued Functions, 655

9. Abelian Integrals and Functions, 668

10. Conformal Mapping, 666

11. The Representation of Functions and Exceptional Values, 667

28. Partial Differential Equations in the Nineteenth Century, 671

1. Introduction, 671

2. The Heat Equation and Fourier Series, 671

3. Closed Solutions; the Fourier Integral, 679

4. The potential Equation and Green,s Theorem, 681

5. Curvilinear Coordinates, 6ST

6. The Wave Equation and the Reduced Wave Equation, 690

7. Systems of Partial Differential Equations, 696

8. Existence Theorems, 699

29. Ordinary Differential Equations in the Nineteenth Century, 709

1l. Introduction, 709

2. Series Solutions and Special Functions, 709

3. Strum- Liouville Theory, 715

4. Existence Theorems, 717

5. The Theory of Singularities, 72!

6. Automorphic Functions, 726

7. Hill's Work on periodic Solutions of Linear Equations, 730

8. Nonlinear Differential Equations: The Qualitative Theory,732

30. The Calculus of Variations in the Nineteenth Century, 739

1. Introduction, 739

2. Mathematical physics and the calculus of variations, 739

3. Mathematical Extensions of the Calculus of Variations proper, 745

4. Related Problems in the Calculus of Variations, 749

31. Galois Theory, 752

1. Introduction, 752

2. Binomial Equations, 752

3. Abel's Work on the Solution of Equations by Radicals, 754

4. Galois's Theory of Solvabitity, 755

5. The Geometric Construction Problems,763

6. The Theory of Substitution Groups,764

32. Quaternions, Vectors, and Linear Associative Algebras,772

1. The Foundation of Algebra on Permanence of Form, 772

2. The Search for a Three-Dimensional " Complex Number," 776

3. The Nature of Quaternions, 779

4. Grassman's Calculus of Extension, 782

5. From Quaternions to Vectors, 785

6. Linear Associative Algebras, 791

33. Determinants and Matrices, 795

1. Introduction, 795

2' Some New Uses of Determinants, 795

3' Determinants and Quadratic Forms, 799

4. Matrices, 804

34. The Theory of Numbers in the Nineteenth Century, Bl3

l. Introduction, 813

2. The Theory of Congruences, 813

3. Algebraic Numbers, 818

4. The Ideals of Dedekind, 822

5. The Theory of Forms, 826 6' Analytic Number Theory, 829

35. The Revival of Projective Geometry, 834

1. The Renewal of Interest in Geometry' 834

2. Synthetic Euclidean Geometry, 837

3. The Revival of Synthetic Projective Geometry, 8'10

4. Algebraic Projective Geometry, 852

5. Higher Plane Curves and Surfaces' 855

36. Non-Euclidean Geometry' 86l

1. Introduction, 861

2. The Status of Euclidean Geometry About 1800,861

3.The Research on the Parallel Axiom,363

4.Foreshadowings of Non-Euclidean Geometry,867

5. The Creation of Non-Euclidean Geometry,869

6. The Technical Content of Non-Euclidean Geometry, 874

7 . The Claims of Lobatchevsky and Bolyai to priority, 877

8. The Implications of Non-Euclidean Geometry, 879

37. The Differential Geometry of Gauss and Riemann, 882

1. Introduction, SS2

2. Gauss's Differential Geometry,882

3. Riemann's Approach to Geometry,SSg

4.The Successors of Riemann,3g6

5. Invariants of Differential Forms, 899

38. Projective and Metric Geometry, 904

1. Introduction, 904

2. Surfaces as Models of Non-Euclidean Geometry,904

3. Projective and Metric Geometry, 906

4. Models and the Consistency Problem, 913

5. Geometry from the Transformation Viewpoint ,9lT

6. The Reality of Non-Euclidean Geometry, 921

39. Algebraic Geometry, 924

1. Background, 924

2. The Theory of Algebraic Invariants, 925

3. The Concept of Birational Transformations, 932

4. The Function-Theoretic Approach to Algebraic Geometry, 934

5. The Uniformization Problem, 937

6. The Algebraic-Geometric Approach,939

7. The Arithmetic Approach,942

8. The Algebraic Geometry of Surfaces, 943

40. The Instillation of Rigor in Analysis, 947

1. Introduction, 947

2. Functions and Their Properties, 949

3. The Derivative,954

4. The Integral,956

5. Infinite Series, 961

6. Fourier Series, 966

7. The Status of Analysis, 972

41 The Foundations of the Real and Transfinite Numbers, 979

1. Introduction, 979

2. Algebraic and Transcendental Numbers, 980

3. The Theory of Irrational Numbers, 982

4. The Theory of Rational Numbers, 987

5. Other Approaches to the Real Number System, 990

6. The Concept of an Infinite Set, 992

7, The Foundation of the Theory of Sets, 994

8. Transfinite Cardinals and Ordinals, 998

9. The Status of Set Theory by 1900, 1002

42. The Foundations of Geometry, 1005

1. The Defects in Euclid, 1005

2. Contributions to the Foundations of Projective Geometry, 1007

3. The Foundations of Euclidean Geometry, l0l0

4. Some Related Foundational Work, l0l5

5. Some Open Questions, l0l7

43. Mathematics as of 1900, 1023

1. The Chief Features of the Nineteenth-Century Developments, 1023

2. The Axiomatic Movement, 1026

3. Mathematics as Man's Creation, 1028

4. The Loss of Truth, 1032

5. Mathematics as the Study of Arbitrary Structures, 1036

6, The Problem of Consistency, 1038

7. A Glance Ahead, 1039

44. The Theory of Functions of Real Variables, 1040

1. The Origins, 1040

2. The Stieltjes Integral, l04l

3. Early Work on Content and Mcaaure, l04l

4. The Lebesgue Integral, 10'14

5. Generalizations, 1050

45. Integral Equations, 1052

1. Introduction, 1052

2. The Beginning of a General Theory, 1056

3. The Work of Hilbert, 1060

4. The Immediate Successors of Hilbert, 1070

5. Extensions of the Theory, 1073

46. Functional Analysis, 1076

1. The Nature of Functional Analysis, 1076

2. The Theory of Functionals, 1077

3. Linear Functional Analysis, l08l

4. The Axiomatization of Hilbert Space, l09l

47. Divergent Series, 1096

1. Introduction, 1096

2. The Informal Uses of Divergent Series, 1098

3. The Formal Theory of Asymptotic Series, ll03

4. Summability, ll09

48. Tensor Analysis and Differential Geometry, I122

1. The Origins of Tensor Analysis, ll22

2. The Notion of a Tensor, ll23

3. Covariant Differentiation, ll27

4. Parallel Displacement, ll30

5. Generalizations of Riemannian Geometry, I 133

49. The Emergence of Abstract Algebra, 1136

1. The Nineteenth-Century Background, ll36

2. Abstract Group Theory, ll37

3. The Abstract Theory of Fields, 1146

4.Rings, 1150

5.Non-Associative Algebras,1153

6. The Range of Abstract Algebra, I 156

50. The Beginnings of Topology,

ll58 l. The Nature of Topology, lt58

2' Point Set Topology, ll59

3' The Beginnings of Combinatorial Topology, l163

4' The Combinatorial Work of Poincard, ll70

5. Combinatorial invariants, 1176

6. Fixed Point Theorems, ll77

7' Generalizations and Extensions, I179

51. The Foundations of Mathematics, I lB2

1. Introduction, ll82

2. The Paradoxes of Set Theory, ll83

3' The Axiomatization of Set Theory, I 185

4. The Rise of Mathematical Logic, 1187

5. The Logistic School' I 192

6. The Intuitionist School, I197

7' The Formalist School, 1203

8. Some Recent Developments, 1208

List of Abbreviations, l2l 3 Index, l2l7

=================================================

A Look Inside

Reviews

Review Quotes

"[Kline] has produced in this enormous book what must be

the definitive history of mathematical thought.

Clearly written and handsomely produced....

[It makes] this abstract discipline come alive with a sense of organic growth;

we see the personalities, the societies, the false starts,

and the patient building up of a body of knowledge now so complex

that it overwhelms the layman....

For anyone with a background and interest in mathematics

it is an important event--

probably the most comprehensive account of mathematical history we have yet had."--Saturday Review

"[Kline] has produced in this enormous book what must be the definitive history

of mathematical thought.

Clearly written and handsomely produced....

[Itmakes] this abstract discipline come alive with a sense of organic growth;

we see the personalities, the societies, the false starts,

and the patient building up of a body of knowledge now so complex

that it overwhelms the layman....

For anyone with a background and interest in mathematics it is an important event --probably the most comprehensive account of mathematical history we have yet had."--

Saturday Review

"The consistently high quality of presentation,

the accuracy, the readable style, and the stress on the conceptual development of mathematics make this volume a most desirable reference."--

Choice

"The consistently high quality of presentation, the accuracy,

the readable style,

and the stress on the conceptual development of mathematics make

this volume a most desirable reference."--Choice

"This is without doubt a book which should be in the library of every institution where mathematics is taught."--Economist

"This is without doubt a book which should be in the library of every institution where mathematics is taught."--Economist

To find out how to look for other reviews, please see our guides to finding book reviews in the Sciences or Social Sciences and Humanities.

More Details

Author := Kline, Morris, 1908-1992

Title := Mathematical thought from ancient to modern times.

Imprint := New York : Oxford University Press, 1972.

[ Library of Congress Class Number := QA21 .K53 ]

UofT at Mississauga Stacks QA21 .K53

ISBN := 0195014960:

Catalogue Key := 2795115

Includes bibliographies.

=================================================

☀► Table of Contents ☀►

1. Mathematics in Mesopotamia, 3

1. Where Did Mathematics Begin? 3

2. Political History in Mesopotamia,4

3. The Number Symbols, 5

4. Arithmetic Operations, 7

5. Babylonian Algebra, 8

6. Babylonian Geometry, 10

7. The Uses of Mathematics in Babylonia, I I

8- Evaluation of Babylonian Mathematics, 13

2. Egyptian Mathematics, 15

1. Background, 15

2. The Arithmetic, 16

3. Algebra and Geometry, 18

4. Egyptian Uses of Mathematics, 21

5. Summary, 22

3. The Creation of Classical Greek Mathematics,24

1. Background, 24

2. The General Sources,25

3. The Major Schools of the Classical Period,27

4. The Ionian School, 28

5. The Pythagoreans, 28

6. The Eleatic School, 34

7. The Sophist School, 37

8. The Platonic School, 42

9. The School of Eudoxus, 48

10. Aristotle and His School, 5l

4. Euclid and Apollonius, 56

1. Introduction, 56

2. The Background of Euclid's Elements,5T

3. The Definitions and Axioms of the Elements,SE

4. Books I to IV of the Elements,60

5. Book V: The Theory of Proportion, 68

6. Book VI : Similar Figures, 73

7. Bools VII, VIII, and IX: The Theory of Numbers, 77

8. Book X: The Classification of Incommensurables, 80

9. Books XI, XII, and XIII: Solid Geometry and the Method of Exhaustion, El

10. The Merits and Defects of the Elements, S6

11. Other Mathematical Works by Euclid,88

12. The Mathematical Work of Apollonius, S9

5. The Alexandrian Greek Period: Geometry and Trigonometry, l0l

1. The Founding of Alexandria, l0l

2. The Character of Alexandrian Greek Mathematics, 103

3. Areas and Volumes in the Work of Archimedes, 105

4. Areas and Volumes in the Work of Heron, 116

5. Some Exceptional Curves, ll7

6. The Creation of Trigonometry, I 19

7. Late Alexandrian Activity in Geometry, 126

6. The Alexandrian Period: The Reemergence of Arithmetic and Algebra, 131

1. The Symbols and Operations of Greek Arithmetic, l3l

2. Arithmetic and Algebra as an Independent Development, 135

7. The Greek Rationalization of Nature, 145

1. The Inspiration for Greek Mathematics, 145

2. The Beginnings of a Rational View of Nature, 146

3. The Development of the Belief in Mathematical Design, 147

4. Greek Mathematical Astronomy 154

5. Geography, 160

6. Mechanics, 162

7. Optics, 166 8. Astrology, 168

8. The Demise of the Greek World, 171

1. A Review of the Greek Achievements, l7l

2. The Limitations of Greek Mathematics, 173

3. The Problems Bequeathed by the Greeks, 176

4. The Demise of the Greek Civilization, 177

9. The Mathematics of the Hindus and Arabs, lB3

1. Early Hindu Mathematics, 183

2. Hindu Arithmetic and Algebra of the Period e.o, 20G-1200, 184

3. Hindu Geometry and Trigonometry of the Period e.o, 200-1200, 188

4. The Arabs, 190

5. Arabic Arithmetic and Algebra, l9l

6. Arabic Geometry and Trigonometry, 195

7. Mathematics circa 1300, 197

10. The Medieval Period in Europe, 200

1. The Beginnings of a European Civilization, 200

2. The Materials Available for Learning, 201

3. The Role of Mathematics in Early Medieval Europe, 202

4. The Stagnation in Mathematics, 203

5. The First Revival of the Greek Works, 205

6. The Revival of Rationalism and Interest in Nature, 206

7. Progress in Mathematics Proper, 209

8. Progress in Physical Science, 2ll 9. Summary,2l3

11. The Renaissance, 21 6

1. Revolutionary Influences in Europe, 216

2. The New Intellectual Outlook, 218

3. The Spread of Learning, 220

4. Humanistic Activity in Mathematics, 221

5. The Clamor for the Reform of Science, 223

6. The Rise of Empiricism, 227

12. Mathematical Contributions in the Renaissance, 231

1. Perspective,23l

2. Geometry Proper, 234

3. Algebra, 236

4. Trigonometry, 237

5. The Major Scientific Progress in the Renaissance, 240

6. Remarks on the Renaissance, 247

13. Arithmetic and Algebra in the Sixteenth and Seventeenth Centuries, 250

1. Introduction, 250

2. The Status of the Number System and Arithmetic, 251

3. Symbolism,259

4. The Solution of Third and Fourth Degree Equations,263

5. The Theory of Equations, 270

6. The Binomial Theorem and Allied Topics, 272

7. The Theory of Numbers, 274

8.The Relationship of Algebra to Geometry, 278

14. The Beginnings of Projective Geometry, 285

1. The Rebirth of Geometry, 285

2. The Problems Raised by the Work on Perspective, 286

3. The Work of Desargues, 288

4. The Work of Pascal and La Hire, 295

5. The Emergence of New Principles, 299

15. Coordinate Geometry, 302

1. The Motivation for Coordinate Geometry, 302

2. The Coordinate Geometry of Fermat,303

3.Rend Descartes,SO4

4.Descartes's Work in Coordinate Geometry' 308

5. Seventeenth'Century Extensions of Coordinate Geometry, 317

6' The Importance of Coordinate Geometry 321

16. The Mathematization of Science, 325

1. Introduction, 325

2. Descartes's Concept of Science, 325

3. Galileo's Approach to Science, 327

4. The Function Concept, 335

17. The Creation of the Calculus, 342

1. The Motivation for the Calculus, 342

2. Early Seventeenth-Century Work on the Calculus, 344

3. The Work of Newton, 356

4. The Work of Leibniz, 370

5. A Comparison of the Work of Newton and Leibniz, 378

6. The Controversy over Priority, 380

7. Some Immediate Additions to the Calculus, 381

8. The Soundness of the Calculus, 383

18. Mathematics as of 1700, 391

1. The Transformation of Mathematics, 391

2. Mathematics and Science, 394

3. Communication Among Mathematicians, 396

4. The Prospects for the Eighteenth Century, 398

19. Calculus in the Eighteenth Century, 400

1. Introduction' 400

2. The Function Concept, '103

3. The Technique of Integration and Complex Quantities, 406

4. Elliptic Integrals, 411

5. Further Special Functions' 422

6. The Calculus of Functions of Several Variables, 425

7. The Attempt to Supply Rigor in the Calculus, 426

20. Infinite Series, 436

1. Introduction, 436

2. Initial Work on Infinite Series, 436

3. The Expansion of Functions, 440

4. The Manipulation of Series, 442

5.Trigonometric Series, 454

6. Continued Fractions, 459

7. The Problem of Convergence and Divergence' 460

21. Ordinary Differential Equations in the Eighteenth Century, 468

1. Motivations, 468

2. First Order Ordinary Differential Equations,4Tl

3. Singular Solutions, 476

4. Second Order Equations and the Riccati Equation' 478

5. Higher Order Equations,484

6. The Method of Series, 'i88

7. Systems of Differential Equations, 490 8. Summary,499

22. Partial Differential Equations in the Eighteenth Century, 502

1. introduction 502

2. The Wave Equation, 503

3. Extensions of the Wave Equation' 515

4. Potential Theory, 522

5. First Order Partial Differential Equation' 531

6. Monge and the Theory of Characteristics , 5gO

7. Monge and Nonlinear Second Order Equations, 538

8. Systems of First Order Partial Differential Equations' 5'fl)

9. The Rise of the Mathematical Subject, 542

23. Analytic and Differential Geometry in the Eighteenth Century, 544

1. Introduction, 544

2. Basic Analytic Geometry, 544

3. Higher plane Curves, 547

4. The Beginnings of Differential Geometry, 5.54

5. plane Curves, 555

6. Space Curves, 557

7. The Theory of Surfaces, 562

8. The Mapping problem, 520

24. The Calculus of Variations in the Eighteenth Century, 573

1. The Initial Problems, 573

2.The Early Work of Euler, 577

3.The principle of Least Action, 579

4. The Methodology of Lagrange, 582

5. Lagrange and Least Action,.587

6. The Second Variation, 589

25. Algebra in the Eighteenth Century, 592

1. Status of the Number System, 592

2. The Theory of Equations, 597

3. Determinants and Elimination Theory, 606

4. The Theory of Numbers, 60g

26. Mathematics as of 1800, 614

1. The Rise of Analysis, 614

2. The Motivation for the Eighteenth-Century Work, 616

3. The Problem of Proof 617

4. The Metaphysical Basis,619

5. The Expansion of Mathematical Activity, 621

6. A Glance Ahead, 623

27. Functions of a Complex Variable, 626

1. Introduction, 626

2. The Beginnings of Complex Function Theory,626

3. The Geometrical Representation of Complex Numbers, 628

4. The Foundation of Complex Function Theory, 632

5. Weierstrass's Approach to Function Theory,642

6. Elliptic Functions, 644

7. Hyper Elliptic Integrals and Abel's Theorem, 651

8. Riemann and Multiple-Valued Functions, 655

9. Abelian Integrals and Functions, 668

10. Conformal Mapping, 666

11. The Representation of Functions and Exceptional Values, 667

28. Partial Differential Equations in the Nineteenth Century, 671

1. Introduction, 671

2. The Heat Equation and Fourier Series, 671

3. Closed Solutions; the Fourier Integral, 679

4. The potential Equation and Green,s Theorem, 681

5. Curvilinear Coordinates, 6ST

6. The Wave Equation and the Reduced Wave Equation, 690

7. Systems of Partial Differential Equations, 696

8. Existence Theorems, 699

29. Ordinary Differential Equations in the Nineteenth Century, 709

1l. Introduction, 709

2. Series Solutions and Special Functions, 709

3. Strum- Liouville Theory, 715

4. Existence Theorems, 717

5. The Theory of Singularities, 72!

6. Automorphic Functions, 726

7. Hill's Work on periodic Solutions of Linear Equations, 730

8. Nonlinear Differential Equations: The Qualitative Theory,732

30. The Calculus of Variations in the Nineteenth Century, 739

1. Introduction, 739

2. Mathematical physics and the calculus of variations, 739

3. Mathematical Extensions of the Calculus of Variations proper, 745

4. Related Problems in the Calculus of Variations, 749

31. Galois Theory, 752

1. Introduction, 752

2. Binomial Equations, 752

3. Abel's Work on the Solution of Equations by Radicals, 754

4. Galois's Theory of Solvabitity, 755

5. The Geometric Construction Problems,763

6. The Theory of Substitution Groups,764

32. Quaternions, Vectors, and Linear Associative Algebras,772

1. The Foundation of Algebra on Permanence of Form, 772

2. The Search for a Three-Dimensional " Complex Number," 776

3. The Nature of Quaternions, 779

4. Grassman's Calculus of Extension, 782

5. From Quaternions to Vectors, 785

6. Linear Associative Algebras, 791

33. Determinants and Matrices, 795

1. Introduction, 795

2' Some New Uses of Determinants, 795

3' Determinants and Quadratic Forms, 799

4. Matrices, 804

34. The Theory of Numbers in the Nineteenth Century, Bl3

l. Introduction, 813

2. The Theory of Congruences, 813

3. Algebraic Numbers, 818

4. The Ideals of Dedekind, 822

5. The Theory of Forms, 826 6' Analytic Number Theory, 829

35. The Revival of Projective Geometry, 834

1. The Renewal of Interest in Geometry' 834

2. Synthetic Euclidean Geometry, 837

3. The Revival of Synthetic Projective Geometry, 8'10

4. Algebraic Projective Geometry, 852

5. Higher Plane Curves and Surfaces' 855

36. Non-Euclidean Geometry' 86l

1. Introduction, 861

2. The Status of Euclidean Geometry About 1800,861

3.The Research on the Parallel Axiom,363

4.Foreshadowings of Non-Euclidean Geometry,867

5. The Creation of Non-Euclidean Geometry,869

6. The Technical Content of Non-Euclidean Geometry, 874

7 . The Claims of Lobatchevsky and Bolyai to priority, 877

8. The Implications of Non-Euclidean Geometry, 879

37. The Differential Geometry of Gauss and Riemann, 882

1. Introduction, SS2

2. Gauss's Differential Geometry,882

3. Riemann's Approach to Geometry,SSg

4.The Successors of Riemann,3g6

5. Invariants of Differential Forms, 899

38. Projective and Metric Geometry, 904

1. Introduction, 904

2. Surfaces as Models of Non-Euclidean Geometry,904

3. Projective and Metric Geometry, 906

4. Models and the Consistency Problem, 913

5. Geometry from the Transformation Viewpoint ,9lT

6. The Reality of Non-Euclidean Geometry, 921

39. Algebraic Geometry, 924

1. Background, 924

2. The Theory of Algebraic Invariants, 925

3. The Concept of Birational Transformations, 932

4. The Function-Theoretic Approach to Algebraic Geometry, 934

5. The Uniformization Problem, 937

6. The Algebraic-Geometric Approach,939

7. The Arithmetic Approach,942

8. The Algebraic Geometry of Surfaces, 943

40. The Instillation of Rigor in Analysis, 947

1. Introduction, 947

2. Functions and Their Properties, 949

3. The Derivative,954

4. The Integral,956

5. Infinite Series, 961

6. Fourier Series, 966

7. The Status of Analysis, 972

41 The Foundations of the Real and Transfinite Numbers, 979

1. Introduction, 979

2. Algebraic and Transcendental Numbers, 980

3. The Theory of Irrational Numbers, 982

4. The Theory of Rational Numbers, 987

5. Other Approaches to the Real Number System, 990

6. The Concept of an Infinite Set, 992

7, The Foundation of the Theory of Sets, 994

8. Transfinite Cardinals and Ordinals, 998

9. The Status of Set Theory by 1900, 1002

42. The Foundations of Geometry, 1005

1. The Defects in Euclid, 1005

2. Contributions to the Foundations of Projective Geometry, 1007

3. The Foundations of Euclidean Geometry, l0l0

4. Some Related Foundational Work, l0l5

5. Some Open Questions, l0l7

43. Mathematics as of 1900, 1023

1. The Chief Features of the Nineteenth-Century Developments, 1023

2. The Axiomatic Movement, 1026

3. Mathematics as Man's Creation, 1028

4. The Loss of Truth, 1032

5. Mathematics as the Study of Arbitrary Structures, 1036

6, The Problem of Consistency, 1038

7. A Glance Ahead, 1039

44. The Theory of Functions of Real Variables, 1040

1. The Origins, 1040

2. The Stieltjes Integral, l04l

3. Early Work on Content and Mcaaure, l04l

4. The Lebesgue Integral, 10'14

5. Generalizations, 1050

45. Integral Equations, 1052

1. Introduction, 1052

2. The Beginning of a General Theory, 1056

3. The Work of Hilbert, 1060

4. The Immediate Successors of Hilbert, 1070

5. Extensions of the Theory, 1073

46. Functional Analysis, 1076

1. The Nature of Functional Analysis, 1076

2. The Theory of Functionals, 1077

3. Linear Functional Analysis, l08l

4. The Axiomatization of Hilbert Space, l09l

47. Divergent Series, 1096

1. Introduction, 1096

2. The Informal Uses of Divergent Series, 1098

3. The Formal Theory of Asymptotic Series, ll03

4. Summability, ll09

48. Tensor Analysis and Differential Geometry, I122

1. The Origins of Tensor Analysis, ll22

2. The Notion of a Tensor, ll23

3. Covariant Differentiation, ll27

4. Parallel Displacement, ll30

5. Generalizations of Riemannian Geometry, I 133

49. The Emergence of Abstract Algebra, 1136

1. The Nineteenth-Century Background, ll36

2. Abstract Group Theory, ll37

3. The Abstract Theory of Fields, 1146

4.Rings, 1150

5.Non-Associative Algebras,1153

6. The Range of Abstract Algebra, I 156

50. The Beginnings of Topology,

ll58 l. The Nature of Topology, lt58

2' Point Set Topology, ll59

3' The Beginnings of Combinatorial Topology, l163

4' The Combinatorial Work of Poincard, ll70

5. Combinatorial invariants, 1176

6. Fixed Point Theorems, ll77

7' Generalizations and Extensions, I179

51. The Foundations of Mathematics, I lB2

1. Introduction, ll82

2. The Paradoxes of Set Theory, ll83

3' The Axiomatization of Set Theory, I 185

4. The Rise of Mathematical Logic, 1187

5. The Logistic School' I 192

6. The Intuitionist School, I197

7' The Formalist School, 1203

8. Some Recent Developments, 1208

List of Abbreviations, l2l 3 Index, l2l7

=================================================

A Look Inside

Reviews

Review Quotes

"[Kline] has produced in this enormous book what must be

the definitive history of mathematical thought.

Clearly written and handsomely produced....

[It makes] this abstract discipline come alive with a sense of organic growth;

we see the personalities, the societies, the false starts,

and the patient building up of a body of knowledge now so complex

that it overwhelms the layman....

For anyone with a background and interest in mathematics

it is an important event--

probably the most comprehensive account of mathematical history we have yet had."--Saturday Review

"[Kline] has produced in this enormous book what must be the definitive history

of mathematical thought.

Clearly written and handsomely produced....

[Itmakes] this abstract discipline come alive with a sense of organic growth;

we see the personalities, the societies, the false starts,

and the patient building up of a body of knowledge now so complex

that it overwhelms the layman....

For anyone with a background and interest in mathematics it is an important event --probably the most comprehensive account of mathematical history we have yet had."--

Saturday Review

"The consistently high quality of presentation,

the accuracy, the readable style, and the stress on the conceptual development of mathematics make this volume a most desirable reference."--

Choice

"The consistently high quality of presentation, the accuracy,

the readable style,

and the stress on the conceptual development of mathematics make

this volume a most desirable reference."--Choice

"This is without doubt a book which should be in the library of every institution where mathematics is taught."--Economist

"This is without doubt a book which should be in the library of every institution where mathematics is taught."--Economist

To find out how to look for other reviews, please see our guides to finding book reviews in the Sciences or Social Sciences and Humanities.

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